First-order mean field games

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The theory of Mean Field Games, introduced among others by Lasry and Lions a little bit more than 10 years ago, is an emerging topic in applied mathematics, and has connections with game theory, fluid mechanics, PDEs, calculus of variations and optimal transport. It describes the evolution of a population, where each agent has to choose the strategy (i.e., a path) which best fits his preferences, but is affected by the others through a global mean field. A differential game is considered, with a continuum of players, all indistinguishable and all negligible. In the most typical case we face a congestion game (agents try to avoid the regions with high concentrations), and we look for a Nash equilibrium, which can be translated into a system of PDEs.

Existence of equilibria for regular MFG The case of non-local dependence in the density, approach via measures on the space of trajectory, proof of existence via the Kakutani fixed-point theorem, recovering the MFG system.

ODEs and PDEs in the space of measures Interpretation of the MFG system as a Pontryagin principle. Value functions in differential games. The Master equation. Convergence of the N-players game to the MFG as $N\to \infty$.